How do I verify the trigonometric identity?

Verify the trigonometric identity:
secx(cscx-2sinx)=cotx-tanxsecx(cscx2sinx)=cotxtanx

4 Answers
Apr 1, 2018

Using the identities:
1/cosx= secx1cosx=secx
1/sinx= cscx1sinx=cscx
1-sin^2x= cos^2x1sin2x=cos2x
sinx/cosx= tanxsinxcosx=tanx
cosx/sinx= cotxcosxsinx=cotx

Start:
secx(cscx-2sinx)= cotx-tanxsecx(cscx2sinx)=cotxtanx

1/cosx(1/sinx-2sinx)=cotx-tanx1cosx(1sinx2sinx)=cotxtanx

1/(sinxcosx)-(2sinx)/cosx=cotx-tanx1sinxcosx2sinxcosx=cotxtanx

1/(sinxcosx)-(2sin^2x)/(cosxsinx)=cotx-tanx1sinxcosx2sin2xcosxsinx=cotxtanx

(1-2sin^2x)/(cosxsinx)=cotx-tanx12sin2xcosxsinx=cotxtanx

(1-sin^2x-sin^2x)/(cosxsinx)=cotx-tanx1sin2xsin2xcosxsinx=cotxtanx

(cos^2x-sin^2x)/(cosxsinx)=cotx-tanxcos2xsin2xcosxsinx=cotxtanx

(cos^2x)/(cosxsinx)-(sin^2x)/(cosxsinx)=cotx-tanxcos2xcosxsinxsin2xcosxsinx=cotxtanx

cosx/sinx-sinx/cosx=cotx-tanxcosxsinxsinxcosx=cotxtanx

cotx-tanx= cotx-tanxcotxtanx=cotxtanx

Apr 2, 2018

See below.

Explanation:

When verifying such identities, it's best to leave one side alone, and begin working with the more complex side. In this case, that would be the left side.

So, recall that secx=1/cosx, cscx=1/sinx:secx=1cosx,cscx=1sinx:

1/cosx(1/sinx-2sinx)=cotx-tanx1cosx(1sinx2sinx)=cotxtanx

Distribute:

1/(cosxsinx)-(2sinx)/cosx=cotx-tanx1cosxsinx2sinxcosx=cotxtanx

Subtract the two expressions on the left side, using cosxsinxcosxsinx as a common denominator.
(1-2sin^2x)/(cosxsinx)=cotx-tanx12sin2xcosxsinx=cotxtanx

Recall the identity cos2x=1-2sin^2x=cos^2x-sin^2x.cos2x=12sin2x=cos2xsin2x. Apply to the numerator:

(cos^2x-sin^2x)/(cosxsinx)=cotx-tanxcos2xsin2xcosxsinx=cotxtanx

Now, split up the fraction:

cos^2x/(cosxsinx)-sin^2x/(cosxsinx)=cotx-tanxcos2xcosxsinxsin2xcosxsinx=cotxtanx

Simplify by cancelling out relevant terms:

cosx/sinx-sinx/cosx=cotx-tanxcosxsinxsinxcosx=cotxtanx

Recalling that cotx=cosx/sinx, tanx=sinx/cosx,cotx=cosxsinx,tanx=sinxcosx, this holds:

cotx-tanx=cotx-tanxcotxtanx=cotxtanx

Apr 2, 2018

Use the definition of the trig functions to rewrite the problem:

secx=1/cosxsecx=1cosx

cscx=1/sinxcscx=1sinx

tanx=sinx/cosxtanx=sinxcosx

cotx=1/tanx=1/(sinx/cosx)=cosx/sinxcotx=1tanx=1sinxcosx=cosxsinx

and also the Pythagorean identity:

cos^2+sin^2x=1cos2+sin2x=1

=>1-sin^2x=cos^2x1sin2x=cos2x

Now, rewrite the problem in terms of sine and cosine. I'll start with the left side and manipulate it until it looks exactly like the right side:

LHS=color(red)secx(color(blue)cscx-2sinx)LHS=secx(cscx2sinx)

color(white)(LHS)=color(red)(1/cosx)(color(blue)(1/sinx)-2sinx)LHS=1cosx(1sinx2sinx)

color(white)(LHS)=color(red)(1/cosx)*color(blue)(1/sinx)-color(red)(1/cosx)*2sinxLHS=1cosx1sinx1cosx2sinx

color(white)(LHS)=1/(cosxsinx)-color(red)(1/cosx)*2sinxLHS=1cosxsinx1cosx2sinx

color(white)(LHS)=1/(cosxsinx)-(2sinx)/cosxLHS=1cosxsinx2sinxcosx

color(white)(LHS)=1/(cosxsinx)-2*sinx/cosxLHS=1cosxsinx2sinxcosx

color(white)(LHS)=1/(cosxsinx)-2*tanxLHS=1cosxsinx2tanx

color(white)(LHS)=1/(cosxsinx)-tanx-tanxLHS=1cosxsinxtanxtanx

color(white)(LHS)=1/(cosxsinx)-sinx/cosx-tanxLHS=1cosxsinxsinxcosxtanx

color(white)(LHS)=1/(cosxsinx)-(sinxcolor(red)(*sinx))/(cosxcolor(red)(*sinx))-tanxLHS=1cosxsinxsinxsinxcosxsinxtanx

color(white)(LHS)=1/(cosxsinx)-sin^2x/(cosxsinx)-tanxLHS=1cosxsinxsin2xcosxsinxtanx

color(white)(LHS)=(1-sin^2x)/(cosxsinx)-tanxLHS=1sin2xcosxsinxtanx

color(white)(LHS)=cos^2x/(cosxsinx)-tanxLHS=cos2xcosxsinxtanx

color(white)(LHS)=cos^color(red)cancelcolor(black)2x/(color(red)cancelcolor(black)cosxsinx)-tanx

color(white)(LHS)=cosx/sinx-tanx

color(white)(LHS)=cotx-tanx

color(white)(LHS)=RHS

The identity is proved. Hope this helped!

Apr 4, 2018

LHS=secx(csc-2sinx)

=1/cosx(1/sinx-2sinx)

=1/(cosxsinx)-(2sinx)/cosx

=(cos^2x+sin^2x)/(cosxsinx)-2tanx

=cos^2x/(cosxsinx)+sin^2x/(cosxsinx)-2tanx

=cosx/sinx+sinx/cosx-2tanx

=cotx+tanx-2tanx

=cotx-tanx=RHS